创建逆旋转矩阵时出错

Question

我需要对 3D 多边形（一组平面的有序顶点）进行三角剖分。为此，我将多边形旋转到 X/Z 平面中，并对旋转的顶点应用剪耳算法。之后，我将顶点旋转回来，得到一个三角多边形。但是，我的轮换似乎有错误。要将我的多边形旋转到 X/Z 平面，我执行以下操作：

首先，我采用多边形的法向量并从中创建一个逆旋转矩阵

    internal static Matrix4 CreateInverseRotationMatrix(Vector3 up) // up is normal vector of polygon
    {
        Vector3 right;
        // take the longer axis and calculate a normal vector
        if (up.X.AbsoluteValue > up.Z.AbsoluteValue)
            right = up.Cross(new Vector3(0, 0, 1));
        else
            right = up.Cross(new Vector3(1, 0, 0));

        // create the third vector of the matrix
        var backward = right.Cross(up);
        // create inverse rotation matrix
        Matrix4<Rational> m = new Matrix4<Rational>(right.X, right.Y, right.Z, 0, up.X, up.Y, up.Z, 0, backward.X, backward.Y, backward.Z, 0, 0, 0, 0, 1);
        return m;
    }

之后，我将每个顶点与逆旋转矩阵相乘：

    private List<Vector2> CreateRotatedVertexList(List<Vertex3> contourVertices, Vector3 up)
    {
        var matrix = Matrix4.CreateInverseRotationMatrix(up);
        List<Vector2> points = new List<Vector2>();
        foreach (var vertex in contourVertices)
        {
            var rotatedVertex = Matrix4.Multiply(matrix, vertex.Vector3);
            var point = new Vector2m(rotatedVertex.X, rotatedVertex.Z);
            points.Add(point);
        }

        return points;
    }

Matrix4.Multiply 看起来像这样：

    public static Vector3 Multiply(Matrix4 m, Vector3 vec4)
    {
        Vector3 res = Vector3.Zero();
        res.X = m.M11 * vec4.X + m.M12 * vec4.Y + m.M13 * vec4.Z + m.M14;
        res.Y = m.M21 * vec4.X + m.M22 * vec4.Y + m.M23 * vec4.Z + m.M24;
        res.Z = m.M31 * vec4.X + m.M32 * vec4.Y + m.M33 * vec4.Z + m.M34;
        var w = m.M41*vec4.X + m.M42*vec4.Y + m.M43*vec4.Z + m.M44;
        return res;
    }

我在 Blender 中绘制了一个示例多边形。例如，多边形 P ={(-1 1 1), (-0.5 1 0.5), (0 1 0), (0.5 0.5 0), (1 0 0), (1 -0.5 0.5), (1 - 1 1)}

并且旋转后的多边形PR = {(0 1 -4) (0,5 1 -2,5) (1 1 -1) (0,5 1 0,5) (0 1 2) (- 1 1 2) (-2 1 2)}。

旋转顶点的y轴分量均为1，所以多边形位于X/Z平面内，没错。但是与未旋转的多边形相比，旋转的多边形的形状看起来明显不同，我不知道为什么。

Answer 1

我忘了标准化基向量。所以应该是：

internal static Matrix4 CreateInverseRotationMatrix(Vector3 up) // up is normal vector of polygon
{
    Vector3 right;
    up = up.Unit();
    // take the longer axis and calculate a normal vector
    if (up.X.AbsoluteValue > up.Z.AbsoluteValue)
        right = up.Cross(new Vector3(0, 0, 1));
    else
        right = up.Cross(new Vector3(1, 0, 0));
    right = right.Unit();
    // create the third vector of the matrix
    var backward = right.Cross(up);
    // create inverse rotation matrix
    Matrix4<Rational> m = new Matrix4<Rational>(right.X, right.Y, right.Z, 0, up.X, up.Y, up.Z, 0, backward.X, backward.Y, backward.Z, 0, 0, 0, 0, 1);
    return m;
}

创建逆旋转矩阵时出错

Bug in creating inverse rotation matrix

c#

algebra

rotation

matrix